diff --git a/src/vsparse/_norm_common.py b/src/vsparse/_norm_common.py
index 4d5a78f..97b2d1a 100644
--- a/src/vsparse/_norm_common.py
+++ b/src/vsparse/_norm_common.py
@@ -234,20 +234,20 @@ def _g_np(x: np.ndarray, g_code: int) -> np.ndarray:
 
 @numba.njit(cache=True, parallel=True)
 def _column_stats_major_is_col(
-    major_ptr, values, value_ptr, indices, row_scale, need_b, need_gstats, g_code
+    major_ptr, values, value_ptr, indices, row_scale, need_b, need_gstats, g_code, n_rows
 ):
-    """Per-column ``gsum`` (raw material for ``b``) and sum/sum-of-squares of ``g(scaled)``.
+    """Per-column ``gsum`` (raw material for ``b``) and the ``g(scaled)`` variance inputs.
 
-    Fused into one pass per column (rather than two separate dispatches):
-    unlike the VCSR scatter passes below, a VCSC column's ``gsum`` depends
-    only on that column's own nonzeros, so it's already final by the time
-    the second (``g``-transform) loop over the same nonzeros needs it --
-    no need to wait for every other column to finish first.
+    Sums cover the stored entries only; :func:`_finish_variance` folds in the
+    implicit zeros. A VCSC column's statistics depend only on its own
+    nonzeros, so all three loops fuse into one pass per column.
     """
     n_major = major_ptr.shape[0] - 1
     gsum = np.ones(n_major, dtype=np.float64)
     col_sum = np.zeros(n_major, dtype=np.float64)
-    col_sumsq = np.zeros(n_major, dtype=np.float64)
+    col_m2 = np.zeros(n_major, dtype=np.float64)
+    col_corr = np.zeros(n_major, dtype=np.float64)
+    col_nnz = np.zeros(n_major, dtype=np.float64)
     for j in numba.prange(n_major):  # ty: ignore[not-iterable]
         gs = 1.0
         if need_b:
@@ -259,17 +259,30 @@ def _column_stats_major_is_col(
             gsum[j] = gs
         if need_gstats and gs > 0.0:
             s0 = 0.0
-            s1 = 0.0
+            count = 0.0
             for u in range(major_ptr[j], major_ptr[j + 1]):
                 v = values[u]
                 for k in range(value_ptr[u], value_ptr[u + 1]):
-                    scaled = v / row_scale[indices[k]] / gs
-                    gy = _g(scaled, g_code)
-                    s0 += gy
-                    s1 += gy * gy
+                    s0 += _g(v / row_scale[indices[k]] / gs, g_code)
+                    count += 1.0
             col_sum[j] = s0
-            col_sumsq[j] = s1
-    return gsum, col_sum, col_sumsq
+            col_nnz[j] = count
+            # Deviations about the mean, not `sumsq - mean ** 2`: for a
+            # column with no spread every deviation is 0 exactly, where the
+            # latter subtracts two numbers of size `mean ** 2` and keeps only
+            # their rounding error.
+            mean = s0 / n_rows if n_rows > 0 else 0.0
+            m2 = 0.0
+            corr = 0.0
+            for u in range(major_ptr[j], major_ptr[j + 1]):
+                v = values[u]
+                for k in range(value_ptr[u], value_ptr[u + 1]):
+                    d = _g(v / row_scale[indices[k]] / gs, g_code) - mean
+                    m2 += d * d
+                    corr += d
+            col_m2[j] = m2
+            col_corr[j] = corr
+    return gsum, col_sum, col_m2, col_corr, col_nnz
 
 
 # -- statistics: major=rows -- scatter-add passes ----------------------------
@@ -297,15 +310,47 @@ def _scaled_col_sums_vcs(major_ptr, values, value_ptr, indices, row_scale, n_col
 def _gstats_col_sums_vcs(
     major_ptr, values, value_ptr, indices, row_scale, gene_scale, g_code, n_cols, nthreads
 ):
+    """Per-column sum of ``g(scaled)`` over stored entries, and how many there were."""
     n_major = major_ptr.shape[0] - 1
     chunk = (n_major + nthreads - 1) // nthreads
     partial_sum = np.zeros((nthreads, n_cols), dtype=np.float64)
-    partial_sumsq = np.zeros((nthreads, n_cols), dtype=np.float64)
+    partial_nnz = np.zeros((nthreads, n_cols), dtype=np.float64)
     for t in numba.prange(nthreads):  # ty: ignore[not-iterable]
         start = t * chunk
         end = min(n_major, start + chunk)
         local_sum = partial_sum[t]
-        local_sumsq = partial_sumsq[t]
+        local_nnz = partial_nnz[t]
+        for i in range(start, end):
+            rs = row_scale[i]
+            for u in range(major_ptr[i], major_ptr[i + 1]):
+                v = values[u]
+                for k in range(value_ptr[u], value_ptr[u + 1]):
+                    c = indices[k]
+                    gs = gene_scale[c]
+                    if gs > 0.0:
+                        local_sum[c] += _g(v / rs / gs, g_code)
+                        local_nnz[c] += 1.0
+    return partial_sum.sum(axis=0), partial_nnz.sum(axis=0)
+
+
+@numba.njit(cache=True, parallel=True)
+def _gstats_col_deviations_vcs(
+    major_ptr, values, value_ptr, indices, row_scale, gene_scale, g_code, col_mean, n_cols, nthreads
+):
+    """Per-column ``sum(y - mean)`` and ``sum((y - mean) ** 2)`` over stored entries.
+
+    A second walk of the values: on a row-major layout a column's mean is not
+    final until every row has been scattered into it.
+    """
+    n_major = major_ptr.shape[0] - 1
+    chunk = (n_major + nthreads - 1) // nthreads
+    partial_m2 = np.zeros((nthreads, n_cols), dtype=np.float64)
+    partial_corr = np.zeros((nthreads, n_cols), dtype=np.float64)
+    for t in numba.prange(nthreads):  # ty: ignore[not-iterable]
+        start = t * chunk
+        end = min(n_major, start + chunk)
+        local_m2 = partial_m2[t]
+        local_corr = partial_corr[t]
         for i in range(start, end):
             rs = row_scale[i]
             for u in range(major_ptr[i], major_ptr[i + 1]):
@@ -314,11 +359,26 @@ def _gstats_col_sums_vcs(
                     c = indices[k]
                     gs = gene_scale[c]
                     if gs > 0.0:
-                        scaled = v / rs / gs
-                        gy = _g(scaled, g_code)
-                        local_sum[c] += gy
-                        local_sumsq[c] += gy * gy
-    return partial_sum.sum(axis=0), partial_sumsq.sum(axis=0)
+                        d = _g(v / rs / gs, g_code) - col_mean[c]
+                        local_m2[c] += d * d
+                        local_corr[c] += d
+    return partial_m2.sum(axis=0), partial_corr.sum(axis=0)
+
+
+def _finish_variance(col_sum, m2_stored, corr_stored, col_nnz, n_rows):
+    """Per-column mean and variance, given either layout's stored-only sums."""
+    if n_rows <= 0:
+        zeros = np.zeros_like(col_sum)
+        return zeros, zeros
+    mean = col_sum / n_rows
+    # Implicit zeros sit at `g(0) == 0`, so each deviates by exactly `-mean`
+    # and their contribution is closed-form rather than iterated.
+    n_zero = n_rows - col_nnz
+    m2 = m2_stored + n_zero * mean**2
+    corr = corr_stored - n_zero * mean
+    # `corr ** 2 / n` is the corrected two-pass term (Chan, Golub & LeVeque):
+    # the deviations are about the computed mean, not the exact one.
+    return mean, np.clip((m2 - corr**2 / n_rows) / n_rows, 0.0, None)
 
 
 # -- full materialization -----------------------------------------------------
@@ -608,6 +668,17 @@ def _compute_row_scale(arr: Any, recipe: Recipe) -> np.ndarray:
     return row_scale
 
 
+def _is_constant_column(variance: np.ndarray, mean: np.ndarray, n_rows: int) -> np.ndarray:
+    """Whether each column's variance is indistinguishable from zero."""
+    # A constant column has no unit-variance scaling to be put on, and
+    # dividing by whatever the arithmetic left behind amplifies rounding noise
+    # into an arbitrary O(1) value. Bound from scikit-learn's
+    # `_is_constant_feature` (Chan, Golub & LeVeque), sized to the error
+    # `_finish_variance` can leave.
+    eps = np.finfo(np.float64).eps
+    return variance <= n_rows * eps * variance + (n_rows * mean * eps) ** 2
+
+
 class NormalizedViewBase:
     """Shared implementation for the normalized VCSC/VCSR views.
 
@@ -658,7 +729,7 @@ def __init__(
         if self._format == "csc":
             # One fused pass per column for both -- see _column_stats_major_is_col.
             if need_b or need_gstats:
-                gene_scale, col_sum, col_sumsq = _column_stats_major_is_col(
+                gene_scale, col_sum, col_m2, col_corr, col_nnz = _column_stats_major_is_col(
                     arr.major_ptr,
                     arr.values,
                     arr.value_ptr,
@@ -667,10 +738,11 @@ def __init__(
                     need_b,
                     need_gstats,
                     self.recipe.g_code,
+                    n_rows,
                 )
             else:
                 gene_scale = np.ones(n_cols, dtype=np.float64)
-                col_sum = col_sumsq = np.zeros(n_cols, dtype=np.float64)
+                col_sum = col_m2 = col_corr = col_nnz = np.zeros(n_cols, dtype=np.float64)
         else:
             # VCSR can't fuse these: gene_scale[c] isn't final until every row
             # has been scattered into it, so the g-transform pass has to wait
@@ -684,7 +756,7 @@ def __init__(
                 gene_scale = np.ones(n_cols, dtype=np.float64)
             if need_gstats:
                 nthreads = numba.get_num_threads()
-                col_sum, col_sumsq = _gstats_col_sums_vcs(
+                kernel_args = (
                     arr.major_ptr,
                     arr.values,
                     arr.value_ptr,
@@ -692,23 +764,27 @@ def __init__(
                     row_scale,
                     gene_scale,
                     self.recipe.g_code,
+                )
+                col_sum, col_nnz = _gstats_col_sums_vcs(*kernel_args, n_cols, nthreads)
+                col_m2, col_corr = _gstats_col_deviations_vcs(
+                    *kernel_args,
+                    col_sum / n_rows if n_rows > 0 else np.zeros(n_cols, dtype=np.float64),
                     n_cols,
                     nthreads,
                 )
             else:
-                col_sum = col_sumsq = np.zeros(n_cols, dtype=np.float64)
+                col_sum = col_m2 = col_corr = col_nnz = np.zeros(n_cols, dtype=np.float64)
         self.gene_scale = gene_scale
 
         if need_gstats:
-            mean = col_sum / n_rows if n_rows > 0 else np.zeros(n_cols, dtype=np.float64)
-            variance = np.clip(
-                col_sumsq / n_rows - mean**2 if n_rows > 0 else np.zeros(n_cols), 0.0, None
-            )
+            mean, variance = _finish_variance(col_sum, col_m2, col_corr, col_nnz, n_rows)
             std = np.sqrt(variance)
             col_mean = mean if self.recipe.center else np.zeros(n_cols, dtype=np.float64)
             if self.recipe.post_scale:
                 with np.errstate(divide="ignore", invalid="ignore"):
-                    col_post_scale = np.where(std > 0.0, 1.0 / std, 1.0)
+                    col_post_scale = np.where(
+                        _is_constant_column(variance, mean, n_rows), 1.0, 1.0 / std
+                    )
             else:
                 col_post_scale = np.ones(n_cols, dtype=np.float64)
         else:
diff --git a/tests/test_property_normalization.py b/tests/test_property_normalization.py
index a7db114..2cd78ab 100644
--- a/tests/test_property_normalization.py
+++ b/tests/test_property_normalization.py
@@ -83,7 +83,13 @@ def _recipe_reference(dense: np.ndarray, recipe: str) -> np.ndarray:
     if recipe in ("scanpy", "pearson"):
         std = g.std(axis=0)
         with np.errstate(divide="ignore", invalid="ignore"):
-            s = np.where(std > 0, 1.0 / std, 1.0)
+            # Constant columns have no unit-variance scaling; same bound as
+            # the library's `_is_constant_column`, over numpy's own std.
+            eps = np.finfo(np.float64).eps
+            n = dense.shape[0]
+            var = std**2
+            constant = var <= n * eps * var + (n * g.mean(axis=0) * eps) ** 2
+            s = np.where(constant, 1.0, 1.0 / std)
     else:
         s = np.ones(dense.shape[1])
 
diff --git a/tests/test_vcs_norm_recipes.py b/tests/test_vcs_norm_recipes.py
index e9a4a3e..51c4ea1 100644
--- a/tests/test_vcs_norm_recipes.py
+++ b/tests/test_vcs_norm_recipes.py
@@ -15,7 +15,7 @@
 import scipy.sparse as sp
 
 from vsparse import RECIPES, Recipe, VCSCAnnData, VCSCArray, VCSRArray
-from vsparse._norm_common import NORM_CACHE_MAXSIZE
+from vsparse._norm_common import NORM_CACHE_MAXSIZE, _is_constant_column
 
 
 @pytest.fixture(params=[VCSCArray, VCSRArray])
@@ -66,7 +66,13 @@ def _reference(dense: np.ndarray, recipe: str) -> np.ndarray:
     if recipe in ("scanpy", "pearson"):
         std = g.std(axis=0)
         with np.errstate(divide="ignore", invalid="ignore"):
-            s = np.where(std > 0, 1.0 / std, 1.0)
+            # Constant columns have no unit-variance scaling; same bound as
+            # the library's `_is_constant_column`, over numpy's own std.
+            eps = np.finfo(np.float64).eps
+            n = dense.shape[0]
+            var = std**2
+            constant = var <= n * eps * var + (n * g.mean(axis=0) * eps) ** 2
+            s = np.where(constant, 1.0, 1.0 / std)
     else:
         s = np.ones(dense.shape[1])
 
@@ -356,3 +362,52 @@ def test_anndata_cache_does_not_pin_a_dropped_view():
     assert ref() is None
     # Still reusable -- from obs/varm/uns if not from the retained statistics.
     assert adata.normalized("scanpy", recalculate=False) is not None
+
+
+@pytest.mark.parametrize("n_rows", [2, 5, 6, 17, 64, 501])
+@pytest.mark.parametrize("value", [1.0, 7.0, 9999.0])
+@pytest.mark.parametrize("recipe", ["scanpy", "pearson"])
+def test_a_column_with_no_variance_centers_to_zero(vcls, n_rows, value, recipe):
+    """Depth normalization flattens a one-column matrix, so centering leaves 0."""
+    dense = np.full((n_rows, 1), value)
+    v = vcls.from_scipy(_scipy_for(vcls, dense))
+    out = v.normalized(recipe).toarray()
+    np.testing.assert_allclose(out, np.zeros_like(dense), atol=1e-12)
+
+
+def test_a_constant_column_does_not_suppress_its_neighbours(vcls):
+    """Zeroing a no-variance column must not touch the columns beside it."""
+    rng = np.random.default_rng(0)
+    dense = rng.integers(1, 50, size=(40, 5)).astype(float)
+    dense[:, 2] = 4.0
+    v = vcls.from_scipy(_scipy_for(vcls, dense))
+    out = v.normalized("pearson").toarray()
+    varying = np.delete(out, 2, axis=1)
+    assert np.abs(varying).max() > 0.5
+
+
+@pytest.mark.parametrize("n_rows", [6, 10_000, 1_300_000])
+def test_small_but_real_variance_is_not_called_constant(n_rows):
+    """A coefficient of variation of 1e-06 is signal, not noise, at any scale here."""
+    mean = np.array([9.21])
+    std = 1e-6 * mean[0]
+    assert not _is_constant_column(np.array([std**2]), mean, n_rows)[0]
+
+
+@pytest.mark.parametrize("n_rows", [6, 10_000, 1_300_000])
+def test_constant_detection_still_catches_a_flat_column(n_rows):
+    """Variance down at the arithmetic's own noise floor is treated as zero."""
+    mean = np.array([9.21])
+    noise = n_rows * np.finfo(np.float64).eps * mean[0]
+    assert _is_constant_column(np.array([(noise * 0.1) ** 2]), mean, n_rows)[0]
+
+
+def test_both_layouts_compute_the_same_variance(dense):
+    """Both layouts must produce the same statistics."""
+    if dense.sum() == 0:
+        pytest.skip("all-zero matrix: median row total is 0")
+    for recipe in ("scanpy", "pearson", "parafac2"):
+        r = VCSRArray.from_scipy(sp.csr_array(dense)).normalized(recipe)
+        c = VCSCArray.from_scipy(sp.csc_array(dense)).normalized(recipe)
+        np.testing.assert_allclose(r.col_post_scale, c.col_post_scale, rtol=1e-12)
+        np.testing.assert_allclose(r.col_mean, c.col_mean, rtol=1e-12, atol=1e-15)